Synthesis of optical multilayer systems using genetic algorithms

Synthesis of optical multilayer systems using genetic algorithms S. Martin, J. Rivory, and M. Schoenauer

Stochastic algorithms are a promising method for the synthesis of optical multilayer systems. A method based on the use of genetic algorithms is described and applied to the design of three very different optical filters. Solutions found by genetic algorithms are refined, and results are compared with those of previous publications.

1.

Introduction

There are many different methods for the design of optical multilayer systems. Most of them are refinement methods: They require a starting design that is not quite satisfactory and adjust it to improve its performance.1 Contrary to these methods, synthesis methods generate their own starting design, which is usually refined afterward. In complex cases in which one cannot guess a possible starting solution, refinement methods are dramatically inefficient. It is therefore of great interest to be able to synthesize a solution even if a further treatment is necessary. For example, some methods based on the inverse Fourier transform or, more recently, on solving a linearly constrained quadratic problem can be used.2–4 The accuracy of the first method depends on the choice of the modulus and the phase of the spectral function; moreover, some manipulations of the refractive index such as scaling and tapering are added for optimization of the process.5 The second method can be used only for the design of antireflection coatings. Furthermore, up to now it has not been possible to say if one of these methods gives the optimal solution of the problem to be solved. This paper presents a way of synthesizing multilayer systems 1which we recently introduced in a previous paper62 by use of recently developed stochastic algorithms, called genetic algorithms 1GA’s2.7

S. Martin and J. Rivory are with the Laboratoire d’Optique des Solides, Universite´ Pierre et Marie Curie, 4 Place Jussieu, 75252 Paris, Cedex 05, France. M. Schoenauer is with the Centre de Mathe´matiques Applique´es, Ecole Polytechnique, FF-91128 Palaiseau, France. Received 3 June 1994; revised manuscript received 17 October 1994. 0003-6935@95@132247-08$06.00@0. r 1995 Optical Society of America.

Section 2 is a short introduction to the basic theory of GA’s. Section 3 presents the particular application of this method to the synthesis of multilayer systems. In Section 4 three very different examples are presented that are compared with those reported in previous publications. Finally, concluding comments are made in Section 5. 2.

Genetic Algorithms

The aim of GA’s is the maximization of some positive function F on a space E. The elements of E are called individuals, and a set of individuals is called a population. The basis of the algorithm is to make a random initial population evolve so that its individuals maximize F. The size of that population remains fixed along evolution 1for further details see Refs. 7 and 82. The evolution of the population can be described as follows: Consider the population at the nth iteration, Pn; several steps are made to get the next population, Pn11, as follows. The first step is the evaluation of the performance of Pn; calculate F1X 2 for each individual X in the population Pn. The second step is selection; individuals in Pn reproduce by generating clones of themselves, giving a new population, Qn. The expectation of the number of clones for each individual X is proportional to F1X 2, whereas the total number of such clones is the fixed size of the population. The third step is recombination; from that population of clones a given percentage of couples 1called the recombination percentage2 is drawn at random. These couples undergo an operator called recombination, which transforms them into two new individuals. The other individuals are left unchanged. The fourth step is mutation; from this population of children a prescribed percentage of individuals 1the mutation percentage2 is drawn at random. These individuals undergo an operator called mutation, which trans1 May 1995 @ Vol. 34, No. 13 @ APPLIED OPTICS

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forms them randomly in the search space. The other individuals are left unchanged. The resulting population is Pn11. The fact that individuals having a higher performance have greater probabilities of reproducing, mimicking a crude parody of Darwinian evolution, is the main point of the algorithm. Other individuals tend to disappear slowly. With recombination one can hope that from two individuals at least a better one 1i.e., with a higher performance2 can be generated. The mutation prevents the algorithm from converging too quickly to a local maximum of F. The practical behavior of the algorithm, of course, greatly depends on these operators, and their design is the most important part of any GA implementation. Although theoretical results of a GA assume convergence to the global maximum of F, the convergence time, which is generally high, should prevent people from using them if they are not necessary. Another important point is that GA’s, because of their stochastic nature, will never in practice find the exact optimal solution when E is continuous. Therefore, when the population seems to be stabilized, it is necessary to use a basic refinement method to improve the solution to find the nearest maximum, which has the theoretical probability 1 of being the global maximum, for a sufficiently large population size and number of generations.9 In practice, if the algorithm is really stabilized then we consider this maximum as the global maximum of F. 3.

Application to the Synthesis of Multilayer Systems

A.

Search Space

Search space E in our problem is chosen to be 2N dimensional, where N is the number of layers in the system to be synthesized. An individual X is written as a 2N dimensional vector X 5 51t1, n12; 1t2, n22; . . . 1tN, nN 26, where ti and ni represent the thickness and the refractive index of the ith layer in the system, respectively. To obtain realistic solutions we must force the refractive indices to satisfy the following constraints: nL # ni # nH, ;i [ 51, N6, where nL and nH correspond to the lowest and highest refractive indices available for the problem. Therefore the space E is 3I, J4N with I 5 3tmin, tmax4 and J 5 3nL, nH4. Here I and J are parameters to be entered before the program is run. Another possibility is to choose a discrete space for J where the different values correspond to available materials, i.e., J 5 3n1, n2, . . . np4. Both search spaces have been used. B.

Genetic Operators

When both I and J are continuous intervals, the recombination and mutation operators are described as follows. For recombination, consider two individuals, X1 and X2; the two parameters of the ith layer of the first child are defined as

ni2 5 a1ni,122 1 11 2 a21ni,222, niti 5 a1ni,1ti,12 1 11 2 a21ni,2ti,22. 2248

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The same formulas are applied for the second child with a8 5 1 2 a. Coefficient a is the same for all layers 1although an option exists where a is variable from layer to layer2 and is taken at random in the interval 320.5, 1.54. For mutation, a random value with a centered Gaussian 1with a given covariance2 probability is added to the parameters of each layer of the individual to be mutated 1for parameter t the mutation is made in optical thickness, i.e., the value of nt is mutated2. In the case of discrete intervals for n, the same operators are applied and the found value is replaced by the nearest admissible value. The function F that measures the performance of the population can take several forms. The basic expression that we used was

F1X2 5

A

1

p

p

o

B

53R1lj2 2 Ropt1lj24@dRj62

j51

21

,

112

where R1lj2 and Ropt1lj2 are the calculated and desired reflection values, respectively, of the system and dRj is the tolerance at wavelength lj, although one can imagine other functions, e.g., F1X2 5 3maxj[51,p60 R1lj2 2 Ropt1lj20421. We point out that this choice is crucial and that the final system depends strongly on F. Our choice was dictated by the wish to compare our results with previous publications, which used the merit function ŒF21. The program has been written in C11 on a Hewlett-Packard Apollo station. The reflection of the multilayer system was calculated by use of the matrix formalism.10 4.

Results

The function F used in all following examples is that given by Eq. 112, with identical tolerances of 1% for all wavelengths. The mutation and recombination percentages are 30% and 60%, respectively. Because it is difficult to find a convergence criterion, we decide to fix the maximum number of generations arbitrarily to 1600. The number of individuals in the population is 100. The refinement method used in all examples is the basic gradient method. When refining systems, if the thickness of a layer happens to be zero we choose to drop this layer and to continue the optimization. The merit function used during refinement is that described above. The coating materials used in all our examples are assumed to be nondispersive and nonabsorbing in the wavelength region of interest. As a last technical remark, we calculate the average reflectance by gradually increasing the number of wavelengths used to compute it until the estimated average reflectance does not change. All calculations are performed under the assumption of normal incidence. A.

Antireflection Coating

First we applied the GA method for the synthesis of a wideband antireflection coating in the far-infrared region. This example has been studied several times before.1,2,11 The target was R 5 0 at 47 equidistant wavelengths in the spectral region of 7.7 # l # 12.3

µm. The incident medium is air and the substrate refractive index is n 5 4. The final goal was to obtain a solution with only two materials, which correspond approximately to germanium 1n 5 4.22 and zinc sulfide 1n 5 2.22. Therefore we chose the research space described by I 5 32.20, 4.204 and J 5 30.01, 1.04, with thicknesses expressed in micrometers; the resulting solution is then transformed into a two-material system that is submitted to the refinement. To obtain solutions comparable with those of previous publications, we limit the total optical thickness of our systems to 32 µm. We do this by multiplying the layer thicknesses by 32@nt, where nt is the actual total optical thickness of the system. Note that this normalization is used in all examples. System A1 of Table 1 is a 20-layer system found by GA’s with the procedure described above. Its twomaterial equivalent and the final solution after refine-

Table 1.

Construction Parameters of Solutions Found by GA for Antireflection Coatings

System A1a Layer Air 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 Substrate o nt 1µm2 Merit function Average reflectance 1%2 aContinuous

System A2b

t 1µm2

n

t 1µm2

n

— 1.0744 0.0558 0.4568 0.4153 0.6128 1.0435 1.0078 0.1449 0.8832 0.1856 0.1581 0.5814 0.5608 0.2187 0.2692 0.3049 0.9393 0.3836 0.2063 0.5235

1 2.200 4.200 4.200 3.026 2.493 2.231 4.092 4.194 2.947 2.955 3.686 2.876 3.177 3.172 2.792 3.922 3.989 3.561 3.515 3.763

— 0.1393 0.5740 0.3805 0.4165 0.0517 0.1488 0.4564 0.2381 0.2549 0.6686 0.3534 0.4056 0.0383 0.3433 0.0345 0.5064 0.3205 0.0759 0.1169 0.2179 0.3898 0.0578 0.4742 0.4854 0.6686

1 2.203 2.200 2.200 4.200 4.188 3.880 2.883 3.592 2.200 2.750 4.159 3.957 4.194 4.134 4.103 3.282 3.617 3.052 3.889 3.306 2.953 3.942 3.001 2.753 3.464

—

4.00 32 0.90 0.82

—

4.00 25 1.10 1.02

System A3c t 1µm2

— 1 1.0946 2.20 0.7213 4.20 0.3149 2.20 0.3180 4.20 0.5336 2.20 0.1744 4.20 0.1233 2.20 0.3180 4.20 0.2129 2.20 0.1412 4.20 0.2812 2.20 0.2649 4.20 0.5799 2.20 0.1001 4.20 0.0719 2.20 0.3180 4.20 0.1718 2.20 0.7476 4.20 0.1032 2.20 0.4154 4.20 0.0752 2.20 0.6898 4.20 0.0783 2.20 0.5191 4.20 0.0718 2.20 0.0728 4.20 0.0833 2.20 0.4555 4.20 0.0915 2.20 0.3178 4.20 0.0165 2.20 — 4.00 32 1.03 0.99

variation of n, total optical thickness 32 µm. variation of n, total optical thickness 25 µm. c Discrete variation of n, total optical thickness 32 µm. bContinuous

n

Table 2.

Two-Material Equivalents of the Systems Found by GA for Antireflection Coatings Presented in Table 1

System B1 Layer

t 1µm2

n

1.0744 0.6527 0.8223 0.0696 0.6881 0.0758 0.3616 0.4687 0.0352 0.4687 0.0181 0.2328 0.4120 0.1768 0.3352 0.1645 0.2628 0.1559 0.3779 0.2304 0.2947 0.1514 0.2608 0.5221 0.0842 0.5420 0.1907 0.3561 0.1564 0.3811 0.0712

1 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20

—

4.00

Air 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 Substrate

System B2 t 1µm2

n

0.1392 0.0001 0.9545 0.4678 0.0304 0.1806 0.3327 0.2118 0.5186 0.0948 0.3509 0.1620 0.0063 0.2293 0.0241 0.2285 0.0419 0.1524 0.0076 0.3284 0.0096 0.1106 0.1811 0.1565 0.1476 0.2064 0.1064 0.1205 0.1372 0.1628 0.2716 0.1074 0.1694 0.1544 0.3506 0.1040 0.2889 0.2493 0.1964 0.1247 —

1 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 4.00

ment are shown in Tables 2 and 3, respectively 1systems B1 and C12. The calculated reflectance in the region of interest for the three systems is presented in Figs. 1, 2, and 3. The average reflection of system A1 is 0.82%. The value of the merit function is 0.9, whereas the best performance in Ref. 1, achieved by the use of two different refinement methods one after the other, is 0.64. Nevertheless, our solution has to be transformed into a two-material system and then refined. The final solution contains 17 layers, and its performance is equivalent to the best result in Refs. 1 and 11. Note that the refinement procedure eliminated layers with thicknesses less than 0.1 µm, with only one exception in systems C1 and C3. To test the good behavior of GA’s, because they are 1 May 1995 @ Vol. 34, No. 13 @ APPLIED OPTICS

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Table 3. Construction Parameters of the Systems Found by GA ATable 1B Transformed into Two-Material Systems ATable 2B, after Refinement by the Gradient Method, for Antireflection Coatings

System C1 Layer Air 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Substrate o nt 1µm2 Merit function Average reflectance 1%2

System C2

t 1µm2

n

t 1µm2

n

— 1.0873 0.6009 0.5333 0.1030 1.2600 0.6627 0.2082 0.2666 1.2431 0.2858 0.1300 0.7600 1.0067 0.1117 0.6517 1.3928 0.0777

1 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20

— 1.0874 0.6081 0.5150 0.1123 1.2527 0.6523 0.1668 0.3147 1.2562 0.2478 0.2699 0.6677 0.7109 0.0975 1.1170 0.2508 0.2383 0.1006

1 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20 2.20 4.20

System C3 t 1µm2

n

— 1 1.0865 2.20 0.6028 4.20 0.5332 2.20 0.1031 4.20 1.2573 2.20 0.6619 4.20 0.2055 2.20 0.2638 4.20 1.2822 2.20 0.2762 4.20 0.1299 2.20 0.7851 4.20 0.8793 2.20 0.1488 4.20 0.5978 2.20 0.9896 4.20 0.0222 2.20 0.4427 4.20 0.1100 2.20 0.1677 4.20 — 4.00 32.1 0.66 0.62

Fig. 2. Antireflection problem over the region 7.7 # l # 12.3 µm for a substrate of index 4, using coating materials with indices between 2.2 and 4.2. Systems B1 1B22 are the two-material equivalent systems of the solutions found by the GA with a 32-µm 125-µm2 total optical thickness and 20 1252 layers.

stochastic algorithms, we made several trials with different initial populations of the same size. The performances of the best individuals in each case were slightly different but were always satisfying. However, this means that the algorithm was not yet

stabilized and that the refined solution obtained may be not the optimal one. The technique we used to transform a graded-index system into a system with only two materials was described by Southwell.12 In this method the optical thickness, nt, of the different layers must verify the condition nt 9 l. When this condition is not verified, the layer must be divided into several layers with the same refractive index until the condition is fulfilled. Of course this increases the number of layers of the resulting system. When synthesizing system B1 1and B32 derived from A1 1and A32, we use a compromise solution between the increase in the number of layers and the degradation in the performance.

Fig. 1. Antireflection problem over the region 7.7 # l # 12.3 µm for a substrate of index 4, using coating materials with indices between 2.2 and 4.2. Systems A1 1A22 correspond to the solutions found by the GA with a 32-µm 125-µm2 total optical thickness and 20 1252 layers. System A3 is a solution synthesized by the GA, with 31 layers, using only two coating materials with indices of 2.2 and 4.2.

Fig. 3. Antireflection problem over the region 7.7 # l # 12.3 µm for a substrate of index 4, using coating materials with indices of 2.2 and 4.2. Systems C1 1C22 are the refined solutions of the systems found by GA with a 32-µm 125-µm2 total optical thickness and 20 1252 layers. System C3 is a solution found by GA, using only two coating materials with indices of 2.2 and 4.2 after refinement by the gradient method.

2250

— 4.00 31.2 0.65 0.62

— 4.00 27.3 0.70 0.66

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Table 4.

Solution Found by GA before AD1B and after AD2B Refinement for Beam Splitters

System D1 Layer Air 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 Substrate o nt 1µm2 Merit function Average reflectance 1%2

System D2

t 1µm2

n

t 1µm2

n

— 0.0609 0.0164 0.0895 0.0498 0.0748 0.0900 0.0314 0.0419 0.0294 0.0906 0.0926 0.0425 0.0306 0.0172 0.0238 0.0479 0.0504 0.0811 0.0158 0.0625 —

1 2.347 1.402 1.350 2.349 2.218 1.607 1.840 1.732 2.309 2.309 2.088 2.223 2.262 2.311 2.273 1.996 1.788 1.543 1.559 1.471 1.52

— 0.0617 0.0172 0.0916 0.0523 0.0740 0.0939 0.0332 0.0456 0.0338 0.0931 0.1015 0.0500 0.0311 0.0182 0.0261 0.0448 0.0436 0.0592 0.0173 0.0549 —

1 2.350 1.350 1.350 2.350 2.226 1.566 1.764 1.678 2.317 2.350 1.953 1.969 2.273 2.281 2.217 2.273 1.944 1.816 1.423 1.615 1.52

2.00 0.41 50.06

2.03 0.06 50.02

The total optical thickness of 32 µm has been commonly used in previous publications; we decided to examine whether the choice of a thinner system would deteriorate the performance. System A2 1Table 12 is a 25-layer system with a 25-µm total optical thickness; its two-material equivalent and the refined solution are systems B2 and C2 1Tables 2 and 32. The average reflectance of refined system C2 is 0.66% and can be compared with that of system C1, which is 0.62%. Its total optical thickness has been slightly increased by the refinement method to 27.3 µm, and its merit function value is 0.70. This performance is satisfactory and even better than that of the system found in Ref. 2, in which antireflection problems are reduced to the solving of a quadratic programming problem with linear constraints. Finally, we try to synthesize directly a two-material system with a 32-µm total optical thickness. Thus in this case J is a discrete interval with only two possible values: 2.20 and 4.20. Because adajcent layers with identical indices constitute in fact only one layer, the effective final number of layers in the system is much smaller than the dimension of the individuals. Therefore we choose 80 for the dimension of the individuals. The solution given by the GA is represented in Table 1 1system A32 and contains 31 layers. The value of the merit function is 1.03. The system obtained after refinement is C3 of Table 3. It is surprising to see that there is almost no difference between its performance and that of system C1. Resulting reflectance curves cannot be dis-

Fig. 4. Beam-splitter problem over the region 0.4 # l # 1.0 µm for a substrate of index 1.52, using coating materials with indices between 1.35 and 2.35. System D1 is a 20-layer solution with a 2-µm total optical thickness synthesized by the GA. System D2 corresponds to system D1 after refinement with the gradient method.

tinguished 1Fig. 32. A comparison between the three systems shows that the first six layers are almost identical. Layers 1 to 12 of systems C1 and C3 are only very slightly different. This would mean that they are essential. This is confirmed by the fact that the best system in Ref. 11 closely resembles system C1. B.

Beam Splitter

The second example concerns the synthesis of a beam splitter in the region 0.4–1.0 µm, also presented in Ref. 1. The target is defined by 50% reflectance at 31 equidistant wavelengths. The best solution given in this paper is already very good 111 layers, total optical thickness of 1.56 µm, merit function equal to 0.282.

Fig. 5. Rejection filter of 90% over the region 0.5 # l # 0.7 µm, with a 40-nm bandwidth centered at 0.6 µm, using coating materials with indices between 1.35 and 2.20 1substrate is glass, n 5 1.522. System E1 is the solution found by GA 120-µm total optical thickness, 40 layers2. System E2 is obtained after refinement of system E1 by the gradient method. 1 May 1995 @ Vol. 34, No. 13 @ APPLIED OPTICS

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Table 5. Construction Parameters of the System Synthesized by GA before AE1B and after AE2B Refinement for Rejection Filters

System E1 Layer Air 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 Substrate

System E2

t 1µm2

n

t 1µm2

n

— 0.1669 0.3389 0.2015 0.3878 0.2509 0.3489 0.2423 0.2969 0.4931 0.3465 0.2257 0.2484 0.4758 0.3746 0.5393 0.2439 0.3178 0.1804 0.1713 0.5426 0.1207 0.2230 0.1702 0.4466 0.3099 0.0623 0.0274 0.3775 0.1829 0.2401 0.2562 0.1313 0.4719 0.5428 0.4973 0.1531 0.1670 0.2186 0.4772 0.1372 —

1 1.350 1.359 1.626 1.358 1.546 1.554 1.699 1.673 1.886 2.200 1.807 2.026 1.523 2.048 1.389 1.774 1.447 2.058 2.197 1.630 1.419 1.968 1.892 1.590 2.030 2.084 1.923 1.663 1.902 2.110 1.976 1.888 1.646 1.887 1.514 1.830 1.994 1.923 1.553 1.685 1.52

— 0.1122 0.2902 0.1943 0.3442 0.2097 0.3041 0.2508 0.3148 0.5159 0.3591 0.2317 0.2296 0.4651 0.3454 0.5551 0.2418 0.2987 0.1721 0.1793 0.6521 0.1177 0.2080 0.1788 0.4168 0.2933 0.0618 0.0254 0.3731 0.1886 0.2321 0.2398 0.1351 0.4654 0.5071 0.4533 0.1374 0.1687 0.2462 0.4438 0.1340 —

1 1.377 1.709 1.883 1.567 1.758 1.631 1.733 1.626 1.802 2.132 1.817 2.140 1.567 2.200 1.350 1.852 1.479 2.109 2.200 1.350 1.350 2.126 1.944 1.629 2.133 2.048 1.966 1.642 1.938 2.134 2.100 1.904 1.698 1.987 1.777 2.068 1.953 1.805 1.692 1.676 1.52

By increasing the number of layers to 20, we believe that the performance could be improved. The total optical thickness is 2 µm, I 5 31.35, 2.354, and J 5 30.05, 0.54. Table 4 gives the optical thickness and refractive index value of each layer in the system found by GA’s before and after refinement 1systems D1 and D22. Once again we can see that the system found by the GA should be refined to obtain the nearest minimum of the merit function. Figure 4 plots the reflectance of both systems. In this case, as in the next example, both the thickness and the refractive index of the layers were varied during refinement. The average reflectance of systems D1 and D2 is 50.06% and 50.02%, respectively. The value of the merit function for the refined solution is 0.06 1compared with 0.28 in Ref. 12. Note that the 2252

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number of layers is in fact 19 and the total optical thickness is almost 2 µm. A two-material equivalent is not presented in this example, nor in the next one. C.

Rejection Filter

In the last example we applied the GA to the synthesis of a rejection filter in the visible region. The incident medium is air and the substrate is glass 1n 5 1.522. The target was defined by a reflection equal to 90% between 0.58 and 0.62 µm and zero outside the band in the region of interest. The merit function was calculated at 21 equidistant wavelengths between 0.5 and 0.7 µm. The total optical thickness is 20 µm, I 5 31.35, 2.204, J 5 30.01, 0.14, and the number of layers is 40. The reflectance of the system found by the GA before 1E12 and after 1E22 refinement is shown in Fig. 5. Note that the deep hole in the reflectance of system E1 in the rejection band is due to the small number of target wavelengths. With more wavelengths this could be avoided. The construction parameters of these systems are given in Table 5. These results can be compared with those obtained by the inverse Fourier transform method.3,13 System F, whose refractive index profile is plotted in Fig. 6, is a rugate filter synthesized by use of the inverse Fourier transform method with optimal phase modulation and has the same total optical thickness as system E1. The reflectance of systems F and E2 is plotted in Fig. 7. The results are not really comparable because for system F the incident medium and the substrate are identical 1glass2. Nevertheless, it can be seen that system E2 is centered at the same wavelength as the target, and that its bandwidth is narrower than that of system F and closer to the target. For completeness, we mention that in the inverse Fourier transform method the half-width of the rejection band can be reduced by the addition of a

Fig. 6. Rejection filter of 90% over the region 0.5 # l # 0.7 µm, with a 40-nm bandwidth centered at 0.6 µm, using coating materials with indices between 1.35 and 2.20 1substrate and incident medium are identical, n 5 1.522. The refractive index profile of system F was synthesized by the inverse Fourier transform method, using optimal phase modulation.

Fig. 7. Rejection filter of 90% over the region 0.5 # l # 0.7 µm, with a 40-nm bandwidth centered at 0.6 µm, using coating materials with indices between 1.35 and 2.20. System E2 is the solution found by GA after refinement. System F is synthesized by the inverse Fourier transform method and has the refractive index profile shown in Fig. 6. The reflectance of system F is calculated with glass as the incident medium.

scaling of the refractive index profile5; this procedure has not been used for system F. The average reflectance of system E2 is 0.30% for 0.500 , l , 0.575 µm and 0.26% for 0.625 , l , 0.700 µm. Between 0.585 and 0.615 µm the average reflectance is 90.07%, with a maximum of 90.95% 1at 0.611 µm2 and a minimum of 88.7% 1at 0.615 µm2. In this case the number of target wavelengths is a crucial parameter: The more wavelengths used for the calculation, the more abrupt the filter slope. During refinement we did not specify any value for the reflectance between 0.675 and 0.685 µm and between 0.615 and 0.625 µm, i.e., at the edges of the reflectance band. We did not investigate the influence of these bounds on the result and do not believe that they are optimal. 5.

Conclusion

We now compare the advantages of the method described here to those of the methods already existing. It should be stressed that because we intended to demonstrate the multiple possibilities of the GA in the design of interference coatings, we did not try to optimize the refinement procedure, and we believe that better systems can be obtained. If there is a point at which GA’s do not look their best, it is without any doubt at the CPU time. When discrete search spaces are used, the convergence time is 1 order of magnitude less than that in continuous spaces.14 Therefore, when a search is made in continuous spaces, time becomes a problem. In our examples the average CPU time 1on Hewlett-Packard 700 workstations2 was 6–10 h. Despite that time, the algorithm is not completely stabilized and a refinement method, even a basic one such as the gradient method, is necessary to reach the nearest local optimum.

Nevertheless, genetic algorithms have proved their efficiency and versatility in synthesizing optical multilayer systems. Because no starting design is necessary, they should be the method to use when no other classical method works. In each example the results are really satisfying and at least as good as the best system ever published. Furthermore, they can also be used to synthesize coatings, including materials with absorption and dispersion. In this case not only the thickness of the layers but also the sequence of materials can be optimized.15 Genetic algorithms, like simulated annealing, are believed to give the optimal solution of a given problem with high probability. As Tikhonravov and Dobrowolski pointed out,2 the term optimal is quite general and can be understood in many ways. Therefore we give our definition of the optimal solution to a specific problem defined by the reflectance of the system to be synthesized, its number of layers, the available refractive indices, and its total optical thickness: The optimal solution is the system whose reflectance is the closest to the target; how far the reflectance curve is from the target is defined with a merit function 1without constraints, the term optimal loses all significance2. With this definition, and in the case of discrete research spaces, it is true that GA’s find the optimal solution. Yet, as the thicknesses of the layers can hardly be sampled, it will be interesting to couple together GA’s and a refinement method. This should speed up convergence. Another objective will be to evaluate the performance of the population at some random wavelengths in the region of interest and to keep some percentage of best individuals. This should prevent strong oscillations in the reflectance and suppress the dependence of the solution on the choice of the target wavelengths. The ultimate step in that direction would be to evolve a population of wavelengths as well, whose fitness would be how they degrade the fitness of the current population of filters, mimicking the coevolution of parasites to improve the quality of the solution.16 References 1. J. A. Dobrowolski and R. A. Kemp, ‘‘Refinement of optical multilayer systems with different optimization procedures,’’ Appl. Opt. 29, 2876–2893 119902. 2. A. V. Tikhonravov and J. A. Dobrowolski, ‘‘Quasi-optimal synthesis for antireflection coatings: a new method,’’ Appl. Opt. 32, 4265–4275 19932. 3. B. G. Bovard, ‘‘Derivation of a matrix describing a rugate dielectric thin film,’’ Appl. Opt. 27, 1998–2005 119882. 4. J. A. Dobrowolski and D. Lowe, ‘‘Optical thin film synthesis program based on the use of Fourier transforms,’’ Appl. Opt. 17, 3039–3050 119782. 5. P. G. Verly, J. A. Dobrowolski, W. J. Wild, and R. L. Burton, ‘‘Synthesis of high rejection filters with the Fourier transform method,’’ Appl. Opt. 28, 2864–2875 119892. 6. S. Martin, J. Rivory, and M. Schoenauer, ‘‘Simulated Darwinian evolution of homogeneous multilayer systems: a new method for optical coatings design,’’ Opt. Commun. 110, 503– 506 119942. 7. D. E. Goldberg, Genetic Algorithms in Search, Optimization and Machine Learning 1Addison-Wesley, Reading, Mass., 19892. 1 May 1995 @ Vol. 34, No. 13 @ APPLIED OPTICS

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8. J. Holland, Adaptation in Natural and Artificial Systems 1U. Michigan Press, Ann Arbor, Mich., 19752. 9. R. Cerf, ‘‘Une the´orie assymptotique des algorithmes ge´ne´tiques,’’ Ph.D. dissertation 1Universite´ de Montpellier, Montpellier, France, 19942. 10. H. A. Macleod, Thin-Film Optical Filters 1Hilger, Bristol, 19862, Chap. 2, pp. 32–40. 11. J. A. Aguilera, J. Aguilera, P. Baumeister, A. Bloom, D. Coursen, J. A. Dobrowolski, F. T. Goldstein, D. E. Gustafson, and R. A. Kemp, ‘‘Antireflection coatings for germanium IR optics: a comparison of numerical design methods,’’ Appl. Opt. 27, 2832–2840 119882. 12. W. H. Southwell, ‘‘Coating design using very thin high- and low-index layers,’’ Appl. Opt. 24, 457–460 119852.

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13. J. Druessel, J. Grantham, and P. Haaland, ‘‘Optimal phase modulation for gradient-index optical filters,’’ Opt. Lett. 18, 1583–1585 119932 14. M. Schoenauer and Z. Wu, ‘‘Conception optimale discre`te de structures par algorithmes ge´ne´tiques,’’ presented at the National Symposium on Structure Calculus, Giens, France, May 1993. 15. T. Eisenhammer, M. Lazarov, M. Leutbecher, U. Scho¨ffel, and R. Sizmann, ‘‘Optimization of interference filters with genetic algorithms applied to silver-based heat mirrors,’’ Appl. Opt. 32, 6310–6315 119932. 16. D. Hillis, ‘‘Co-evolving parasites improved simulated evolution as an optimisation procedure,’’ Physica 1D2 42, 228–234 119902.